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Existence and Stability of Travelling Front Solutions forGeneral Auto-catalytic Chemical Reaction Systems

Published online by Cambridge University Press:  12 June 2013

Y. Li  and
Y. Wu
Y. Li
Affiliation:
Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, P.R. China and Department of Mathematics and Statistics, Wright State University, Dayton, OH45435, USA
Y. Wu*
Affiliation:
College of Mathematical Sciences, Capital Normal University, Beijing, 100048, P.R. China
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

This paper is concerned with the existence and stability of travelling front solutionsfor more general autocatalytic chemical reaction systems ut = duxx − uf(v), vt = vxx + uf(v)with d > 0 and d ≠ 1, wheref(v) has super-linear or linear degeneracy atv = 0. By applying Lyapunov-Schmidt decomposition method in someappropriate exponentially weighted spaces, we obtain the existence and continuousdependence of wave fronts with some critical speeds and with exponential spatial decay ford near 1. By applying special phase plane analysis and approximatecenter manifold theorem, the existence of traveling waves with algebraic spatial decay orwith some lower exponential decay is also obtained for d > 0. Further,by spectral estimates and Evans function method, the wave fronts with exponential spatialdecay are proved to be spectrally or linearly stable in some suitable exponentiallyweighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to belocally exponentially stable in some exponentially weighted spaces and Lyapunov stable inCunif(ℝ) space, if the initial perturbation of the waves issmall in both the weighted and unweighted norms; the perturbation of the waves also stayssmall in L1(ℝ) norm and decays algebraically inCunif(ℝ) norm, if the initial perturbation is in additionsmall in L1 norm.

Keywords

travelling wavesexistencestabilityspectral analysis
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 104 - 132
Copyright
© EDP Sciences, 2013

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